In this paper we offer new representations for Drazin inverse of block matrix, which recover some representations from current literature on this subject. 2000 Mathematics Subject Classification: 15A09

Let A be a complex unital Banach algebra with unit 1. The sets of all invertible and quasinilpotent elements (σ(a) = {0}) of A will be denoted by A and A, respectively. The group inverse of a ∈ A is the unique element a ∈ A which satisfies aaa = a, aaa = a, aa = aa. If the group inverse of a exists, a is group invertible. Denote by A the set of all group invertible elements of A. The generalize...

Several representations of the generalized Drazin inverse of a block matrix with a group invertible generalized Schur complement in Banach algebra are presented.

Given an n × n singular matrix A with Ind(A) = k, an index splitting of A is one of the form A = U − V , where R(U) = R(Ak) and N(U) = N(Ak). This splitting, introduced by the first author, generalizes the proper splitting proposed by Berman and Plemmons. Regarding singular systems Au = f , the first author has shown that the iterations u(i+1) = U#V u(i) + U#f converge to ADf , the Drazin inver...

In this paper , some results about the index of matrix and Drazin inverse are given. We then explain applications of them in solving singular linear system of equations.

Given a bounded operator A on a Banach space X with Drazin inverse A and index r, we study the class of group invertible bounded operators B such that / + A (B — A) is invertible and 1Z(B) f]N(A) = {0}. We show that they can be written with respect to the decomposition X = lZ(A) 0 Af(A) as a matrix operator, B = ( _ _ __i _ ), where B\ and B\ + B\9#?1 a r e invertible. ^ if21 t>2\tf\ i*\2 Sever...